A product of cyclic ratio sums is at least nine halves

Proved

WorkbookSource.base_25978

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers . Prove that

\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\left(\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\right) \ge\frac{9}{2}

Source: InternLM Lean-Workbook, record lean_workbook_25978 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_25978 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / b + b / c + c / a) * (a / (a + b) + b / (b + c) + c / (c + a)) ≥ 9 / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_25978; Apache-2.0

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